Fredrik said:
Even if we replace ##\land## with ##\Rightarrow##, all we have there is a statement about the set x, so it can't by itself define the set ##\bigcup_{i\in I} A_i##. It says that for all ##i\in I##, we have ##x\in A_i##. This means that x is an element of every ##A_i## with ##i\in I##. A statement like that must be a part of the definition, but it can't be the whole thing.
Hmmm , by stating that ''This means that x is an element of every ##A_i## with ##i\in I##'' , you are saying that we're making a statement about the set x and not about ##\bigcup_{i\in I} A_i##.
I somewhat intuitively understand but still have some confusion.I will take the liberty of creating a scenario to put things into perspective as I think it might help me in that case.Take note that I am thinking as I type the post.
Suppose a room full of people that have either blond , brown or white hair.There's no bald people or people that have another hair color than blond , brown and white.
Set ##A_1## is made of all people in the room with blond hair , set ##A_2## is made of all people in the room with brown hair and set ##A_3## is made of all people in the room with white hair.
Set ##I = \{1,2,3\}##
A person with blond hair ##\in A_1## but a person with brown hair ##\not \in A_1## and so on...
We can conclude that the set:
$$\bigcup_{i\in I}A_i = A_1 \cup A_2 \cup A_3$$
...is made of every person in the room.
The earlier statement ''This means that x is an element of every ##A_i## with ##i\in I##'' would mean ##1,2,3 \in I## and ##x \in A_i##.If he is in ##A_i## and ##i \in I## then ##x \in A_1## , ##x \in A_2## or ##x \in A_3## , but since he either has blond , brown or white hair , how could he be an element of each ##A_i##? In a way , to find an element of the big set you have to find what's common between elements from the 3 subsets? Like ''being in the room'' in my previous exemple?
Set notation where x = a person in the room: ##\{ x\in\bigcup_{i\in I}A_i : \forall i\in I \land x \in A_i\}##
Does this really define the set of all people in the room?
There are only two standard notations,
$$\bigcup_{i\in I}A_i,$$ and $$\bigcup_{i=1}^n A_i.$$
ok good
I'm not sure what you've been trying to do, but yes, if you want to assign a meaning to the symbol I, you have to do it using one of the standard ways to specify a set. In this case, ##I=\{1,2,3\}## is the obvious option, and ##I=\{n\in\mathbb Z\,|\,1\leq n\leq 3\}## is one of the alternatives.
Ok , so if ##I = \{1,2,3,4,5,6,7,8,9\}## then ##I = \{n \in\mathbb Z\,|\,1\leq n\leq 9\}##?
Lose the 3 on top, and the notation is fine.
Ok so we only use the number or symbol on top with we have ##i=1## at the bottom and not ##i \in I## ?
It doesn't really describe it. It's just an alternative notation for it. (And I think you meant ##A_{1} \cup A_{2} \cup A_{3}##).
What do you mean? I can't write it between {}?
B' is only defined when all the sets we're working with are subsets of some set X. Then B' is defined as X-B. B' is not defined here.
Ok , so B' doesn't really exist except in naive set theory?
No, but you could say e.g. that ##A = \{x\in C~\vert~x\not \in B\}##, if you have already specified what C is.
So I can't say ##A = \{x \in A~\vert~x\not \in B\}## ?
Because of:
Right, you can't use the definition while you're stating it. But you also need to be careful when you use the notation ##\{x\,|\,P(x)\}##, because unlike ##\{x\in y\,|\, P(x)\}##, it's not guaranteed by the axioms to always make sense.
thanks!